Abstracts Subgroups

This course is an introduction to acylindrically hyperbolic groups through the lens of the special class of hyperbolically embedded subgroups. We will begin with definitions and examples, and then we will investigate some nice geometric (and algebraic) properties enjoyed by these subgroups. We will explore connections with random walks, extending actions, extending quasicocycles, and relative hyperbolicity. Hyperbolically embedded subgroups, particularly those that are virtually cyclic, are a powerful tool for proving important results about acylindrically hyperbolic groups. I will introduce some of these results and sketch the ways in which hyperbolically embeddded subgroups are used in their proof. The course will assume no prior knowledge of any of these topics.

This mini-course will be an introduction to the Bestvina-Brady Morse theory. Let G be a group acting geometrically on a polyhedral complex X. Suppose that φ:G -> Z is an epimorphism, and X admits a φ-equivarient map to a line. The Bestvina-Brady Morse theory relates the finiteness properties of the kernel of φ to topological properties of the link of a vertex in X. We will focus on the applications of the theory in the context of right-angled Artin and Coxeter groups.

After a brief introduction to the general theory of homology of groups we will define l^2 homology via actions on simplicial complexes. In the second lecture we will compute several examples of these homology groups. Finally, we will discuss consequences that are of interest to group theorists particularly in the case that these invariants vanish.

I’ll talk about various ways to think about quasiconvexity in hyperbolic and relatively hyperbolic groups, and explain why and when quasiconvexity persists under Dehn filling.

Given an automorphism of a finitely generate free group, one can construct a corresponding semi-direct product of the free group and the integers. The semi-direct product, also known as a free-by-cyclic group, is finitely generated and we can consider its Cayley graph (or some other geometric model you prefer). We are interested in the dynamics of the defining automorphism, the algebraic properties of the free-by-cyclic group, the coarse geometry of the geometric models, and especially the interplay between these. These ideas are heavily influenced by Thurston’s work on surface homeomorphisms and 3-manifolds. This mini-course will give an incomplete survey of the various results (from old to recent) along these lines and introduce some of the key ideas behind them. There will be no shortage of open problems to take home with you.

We’ll explore the problem of finding ‘nice’ models for the classifying spaces of certain quotients of fundamental groups of non-positively curved cube complexes, and we’ll discuss the framework that provides the necessary tools to do so – cubical small-cancellation theory.

Let GGG be a free product G=A1∗A2∗⋯∗Ak∗FNG = A_1 * A_2* \cdots * A_k * F_NG=A1​∗A2​∗⋯∗Ak​∗FN​, where each of the groups AiA_iAi​ is torsion-free and where FNF_NFN​ is a free group of rank NNN, and let O\mathcal{O}O be the associated deformation space. We show that the diameter of the projection of the subset of O\mathcal{O}O where a given element has bounded length to the Z\mathcal{Z}Z–factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of GGG as a hyperbolic group relative to the collection of subgroups AiA_iAi​ together with a given non-peripheral cyclic subgroup. In a future paper, we will apply this theorem to study the geometry of free group extensions. This is joint work with Caglar Uyanik.

A group G satisfies the Minkowski separation if if has a finite quotient Q in which all finite order outer-automorphisms of G descend as automorphisms of Q and are non trivial in Out(Q). This property was identified as useful in order to compare two relatively hyperbolic groups whose JSJ decomposition is given. Applied to certain subgroups of mapping tori (F \rtimes Z) of a free group F, it allows to compare the conjugacy classes of certain automorphisms of F. In this talk, the groups of interest are the subgroups of suspensions by multi-Dehn twists. (Joint work with N. Touikan.)

The distortion function of a subgroup measures the extent to which the intrinsic word metric of the subgroup differs from the metric induced by the ambient group. Ol’shanskii showed that there are almost no restrictions on which functions arise as distortion functions of subgroups of finitely presented groups. This prompts one to ask what happens if one forces the ambient group to be particularly nice, say, for example, to be hyperbolic. I will survey which functions are known to be distortion functions of subgroups of hyperbolic groups and describe joint work with Tim Riley in which we construct new examples of such functions.

A rational number p/q determines a simple closed curve on a once-punctured torus. When the torus is endowed with a complete hyperbolic metric, each rational gets a well-defined length. If the metric is chosen so that the torus is “modular” (that is, when its holonomy group is conjugate into PSL(2,Z)), the lengths of the curves have special arithmetic significance, with connections to Diophantine approximation and number theory. Taking inspiration from McShane’s elegant proof of Aigner’s conjectures, concerning the (partial) ordering of the rationals induced by hyperbolic length on the modular torus, I will describe how hyperbolic geometry can be used to characterize monotonicity of the order so obtained along lines of varying slope in the (q,p)-plane.

Einstein and I introduced relatively geometric actions of relatively hyperbolic groups and instigated the study of relatively geometric actions on CAT(0) cube complexes. I will introduce these actions, give examples and non-examples, and explain how one gets separability of certain quasi-convex subgroups under appropriate hypotheses. I’ll compare the theory of relatively geometric actions to that of geometric actions of hyperbolic groups. This is mostly joint work with Teddy Einstein, and some joint work with Corey Bregman and Kejia Zhu.

One of the first major results about Gromov hyperbolic groups states that the boundary of any one-ended hyperbolic group is always locally connected. This deep result—established in the 1990s by Bestvina-Mess, Levitt, Bowditch, and Swarup—led to advances in JSJ decompositions, boundary classification problems, and the analysis of boundaries. Shortly afterward, Bowditch introduced a natural boundary for any relatively hyperbolic group pair and conjectured that this boundary is always locally connected. However, he established this fact only in the presence of certain awkward and restrictive hypotheses.

In fact, the boundary of every one-ended relatively hyperbolic group pair is locally connected, with no restrictions. This result applies in a general setting, in which the groups in question need not be finitely generated or even countable. (Joint work with Ashani Dasgupta.)

We show that isomorphic finite index subgroups of non-elementary hyperbolic groups must have the same index. In this talk I will present the tools and ideas of the proof.

(Joint work with Andrei Jaikin-Zapirain.) A group is said to be coherent if all of its finitely generated subgroups are finitely presented. In this talk I will sketch a proof of Baumslag’s well known conjecture that all one-relator groups are coherent and discuss applications of the ingredients that go into the proof.

A group is virtually free when it has a free subgroup of finite index. A theorem of Karrass, Pietrowski and Solitar says that equivalently a group is finitely generated and virtually free when it acts on a tree with finite stabilizers and finite quotient. We will introduce the deformation space of these trees, called Outer Space, and relate it to the Outer Space of the free subgroup of finite index.

We show that being hyperbolic is not a hereditary property for groups, not even if we consider subgroups with the strongest possibile finiteness assumption: a hyperbolic group may contain a finite type subgroup that is not itself hyperbolic. We show how to construct such examples from a hyperbolic 5-manifold that fibers over the circle.

We start with a Gromov-hyperbolic surface bundle over a graph, and drill out essential simple closed curves from fibers. We prove that the resulting drilled bundles have relatively hyperbolic fundamental group with peripheral Z+Z groups. We then gives reasonably mild sufficient conditions to guarantee cubulability of these groups. This is joint work with Biswajit Nag.

Following C.T.C. Wall, we say that a group G is of type F_n if it admits a classifying space which is a CW-complex with finite n-skeleton. For n=2, one recovers the notion of being finitely presented. We prove that in a cocompact arithmetic lattice of the group PU(m,1), with positive first Betti number, deep enough finite index subgroups admit plenty of homomorphisms to Z with kernel of type F_{m-1} but not of type F_m. This provides many non-hyperbolic finitely presented subgroups of hyperbolic groups and answers an old question of Brady. This is based on a joint work with C. Llosa Isenrich.

Hyperbolization procedures are constructions that turn polyhedra into spaces of negative curvature, while preserving some topological features. They have been used to construct examples of manifolds that exhibit various pathologies, despite having negative curvature. One may expect the fundamental groups of these manifolds to be pathological as well. On the other hand, in joint work with J. Lafont, we show that these groups admit nice actions on CAT(0) cube complexes. As an application, we obtain new examples of negatively curved Riemannian manifolds whose fundamental groups are virtually special and algebraically fibered.

The Dehn fillings of a relatively hyperbolic group are useful relatively hyperbolic quotients constructed in a way inspired by Thurston’s hyperbolic Dehn filling theorem. In the context of mapping class groups, a reasonable analogue of Dehn fillings are quotients by large powers of Dehn twists. I will discuss the algebra and geometry of these and related quotients.

Rigidity problems in geometric group theory frequently have the following form: if two finitely generated groups share a geometric structure, do they share algebraic structure? We consider two finitely generated groups that are either quasi-isometric or act geometrically on the same proper metric space, and ask if they are virtually isomorphic. The work of Papasoglu–Whyte demonstrates that infinite-ended groups are quasi-isometrically flexible, but our results show that if you assume a common geometric model, then there is often rigidity. We introduce the notion of graphical discreteness and prove rigidity results for many graphs of groups with graphically discrete vertex groups. For example, we prove action rigidity for many free products of graphically discrete groups. We also show quasi-isometric rigidity of an amalgamated free product of closed hyperbolic 3-manifold groups over quasi-convex, almost malnormal nonabelian free subgroups. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

We show that every atoroidal, endperiodic map of an infinite-type surface is isotopic to a homeomorphism that is naturally the first return map of a pseudo-Anosov suspension flow on a fibered manifold. Morally, these maps are all obtained by “spinning” fibers around a surfaces in the boundary of the fibered cone. The structure associated to these spun pseudo-Anosov maps allows for several applications. These include defining and characterizing stretch factors of endperiodic maps, relating Cantwell—Conlon foliation cones to Thurston’s fibered cones, and defining a convex entropy function on these cones that extends log(stretch factor). This is joint work with Michael Landry and Yair Minsky.

I will report on technology and perspectives used to solve the conjugacy problem in Out(F_3). This is joint work with Armando Martino, François Dahmani, and Stefano Francaviglia.

We discuss how finiteness properties of algebraic fibers of polyfree groups can be used to study coherence of the outer automorphism group of a free group.

I will discuss some new invariants of 2-complexes. They are inspired by recent developments in the theory of one-relator groups, but also have the potential to unify the theories of many well-studied families including small-cancellation presentation complexes, CAT(0) 2-complexes and 3-manifold spines, in addition to the motivating examples of one-relator presentation complexes. The fundamental result is that these invariants are the extrema of explicit linear-programming problems, and in particular are rational, computable and realised. The definitions suggest a conjectural “map” of 2-complexes, which I will attempt to describe.